The theory of stochastic optimal control deals with the following problem:

Find $\quad\sup\limits_{u} \; \mathrm E[g(X^{(u,x)}_T)]$

where $X^{(u,x)}_t$ solves the following controlled SDE:

$dX_t=\sigma(u,X_t,t)dW$

$X_0 = x$ and $u(\omega,t)$ is the (adapted) control.

**In my case:**

$u(\omega,t)\in U$ for some convex set $U\subset\mathbb R $

and $g$ is a convex function!

**My problem:**

Most of the literature about stochastic optimal control, esp. maximum principles, deals with concave functions $g$ so that most of it does not apply to the convex case.

Is it, because the convex case the easier one and I just do not see how/why?

Where can I find something about the convex case?

Thanks,

Johannes